# THE IDEAL-GAS EQUATION OF STATE

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"THE IDEAL-GAS EQUATION OF STATE"

```					    THE IDEAL-GAS EQUATION OF STATE
•   Equation of state: Any equation that relates the pressure, temperature,
and specific volume of a substance.
•   The simplest and best-known equation of state for substances in the gas
phase is the ideal-gas equation of state. This equation predicts the P-v-T
behavior of a gas quite accurately within some properly selected region.

Ideal gas equation
of state

Different substances have different
gas constants.                    1
Mass = Molar mass  Mole number                Ideal gas equation at two
states for a fixed mass

Real gases
behave as an ideal
Various
gas at low
NR                NR T   expressions densities (i.e., low
of ideal gas pressure, high
equation     temperature).
RT

The ideal-gas
relation often is not
applicable to real
gases; thus, care
should be exercised
when using it.
Properties per
unit mole are
denoted with a
bar on the top.                                     2
Is Water Vapor an Ideal Gas?
•   At pressures below 10 kPa, water
vapor can be treated as an ideal
gas, regardless of its temperature,
with negligible error (less than 0.1
percent).
•   At higher pressures, however, the
ideal gas assumption yields
unacceptable errors, particularly in
the vicinity of the critical point and
the saturated vapor line.
•   In air-conditioning applications, the
water vapor in the air can be
treated as an ideal gas. Why?
•   In steam power plant applications,
however, the pressures involved
are usually very high; therefore,
ideal-gas relations should not be
used.

Percentage of error ([|vtable - videal|/vtable] 100) involved in
assuming steam to be an ideal gas, and the region where steam
3
can be treated as an ideal gas with less than 1 percent error.
COMPRESSIBILITY FACTOR—A MEASURE
OF DEVIATION FROM IDEAL-GAS BEHAVIOR
Compressibility factor Z        The farther away Z is from unity, the more the
A factor that accounts for      gas deviates from ideal-gas behavior.
the deviation of real gases     Gases behave as an ideal gas at low densities
from ideal-gas behavior at      (i.e., low pressure, high temperature).
a given temperature and         Question: What is the criteria for low pressure
pressure.                       and high temperature?
Answer: The pressure or temperature of a gas
is high or low relative to its critical temperature
or pressure.

At very low pressures, all gases approach
The compressibility factor is     ideal-gas behavior (regardless of their
unity for ideal gases.            temperature).                                 4
Reduced        Reduced
pressure       temperature
Pseudo-reduced            Z can also be determined from
specific volume               a knowledge of PR and vR.

Gases deviate from the
ideal-gas behavior the
most in the neighborhood
of the critical point. 5
Comparison of Z factors for various gases.
Ideal Gas Model

pv  RT
pR  1, TR  1
u  u (T ) only
h  h(T )  u (T )  RT

Do not be confused with superheated water vapor!
INTERNAL ENERGY, ENTHALPY,
AND SPECIFIC HEATS OF IDEAL GASES

Joule showed
using this
experimental                         Internal energy and
apparatus that   For ideal gases,
enthalpy change of
u=u(T)           u, h, cv, and cp
an ideal gas
vary with
temperature only.                          7
•   At low pressures, all real gases approach     •    u and h data for a number of
ideal-gas behavior, and therefore their            gases have been tabulated.
specific heats depend on temperature only.    •    These tables are obtained by
•   The specific heats of real gases at low            choosing an arbitrary reference
pressures are called ideal-gas specific            point and performing the
heats, or zero-pressure specific heats, and        integrations by treating state 1
are often denoted cp0 and cv0.                     as the reference state.

Ideal-gas
constant-
pressure
specific heats
for some
gases (see           In the preparation of ideal-gas
Table A–2c           tables, 0 K is chosen as the
for cp               reference temperature.
8
equations).
Internal energy and enthalpy change when
specific heat is taken constant at an
average value

(kJ/kg)

For small temperature intervals, the
specific heats may be assumed to vary
linearly with temperature.

The relation  u = cv T
is valid for any kind of
process, constant-
volume or not.                               9
Three ways of calculating u and h
1. By using the tabulated u and h data.
This is the easiest and most
accurate way when tables are
2. By using the cv or cp relations (Table
A-2c) as a function of temperature
and performing the integrations. This
is very inconvenient for hand
calculations but quite desirable for
computerized calculations. The
results obtained are very accurate.
3. By using average specific heats.
This is very simple and certainly very
convenient when property tables are
not available. The results obtained
are reasonably accurate if the           Three ways of calculating u.
temperature interval is not very
large.

10
Specific Heat Relations of Ideal Gases
The relationship between cp, cv and R

dh = cpdT and du = cvdT               On a molar basis

Specific
heat ratio

•    The specific ratio varies with
temperature, but this variation is
very mild.
•    For monatomic gases (helium,
argon, etc.), its value is essentially
constant at 1.667.
The cp of an ideal gas can be
determined from a knowledge of    •    Many diatomic gases, including air,
cv and R.                              have a specific heat ratio of about
1.4 at room temperature.
11
Internal Energy Changes of incompressible substance

Enthalpy Changes

The enthalpy of a
compressed liquid
A more accurate relation than                       12
EX 3.8 One pound of air in a piston-cylinder assembly undergoes a
thermodynamic cycle consisting of three processes.
Process 1–2: Constant specific volume
Process 2–3: Constant-temperature expansion
Process 3–1: Constant-pressure compression
At state 1, the temperature is 540°R, and the pressure is 1 atm. At state 2, the
pressure is 2 atm. Employing the ideal gas equation of state,
(a) sketch the cycle on p–v coordinates.
(b) determine the temperature at state 2, in °R.
(c) determine the specific volume at state 3, in ft3/lb.

Power cycle or Refrigeration cycle?
Using Ideal Gas Tables
A piston-cylinder assembly contains 2 lb of air at a temperature
of 540 oR and a pressure of 1 atm. The air compressed to a
state where the temperature is 840 oR and the pressure is 6
atm. During the compression, there is a heat transfer from the
air to the surroundings equal to 20 Btu. Using the ideal gas
model for air, determine the work during the process in Btu.
690 oR = 230 oF
Specific heat functions

cp
   T  T 2  T 3  T 4   h2-h1?
R
Using constant c
u (T2 )  u(T1 )  cv (T2  T1 )
h(T2 )  h(T1 )  c p (T2  T1 )
Two tanks are connected by a valve. One tank contains 2 kg of carbon
monoxide gas at 77 oC and 0.7 bar. The other tank holds 8 kg of the same
gas at 27 oC and 1.2 bar. The valve is opened and the gases are allowed to
mix while receiving energy by heat transfer from the surroundings. The final
equilibirium temperature is 42 oC. Using the ideal gas model with constant
cv, determine (a) the final equilibrium pressure, in bar (b) the heat transfer
for the process, in kJ.
Polytropic, Isothermal, and Isobaric processes
Polytropic process: C, n (polytropic exponent) constants
Polytropic
process
n 1
n 1
T2  V1            P      n
Polytropic and for ideal gas                     2 
T1  V2 
 
P
 1
When n = 1
(isothermal process)

Constant pressure process

What is the boundary
work for a constant-
volume process?

Schematic and
P-V diagram for
a polytropic
19
process.
Ex. 3.12
• Air undergoes a polytropic compression in a piston–cylinder
assembly from p1=1 atm, T1=70 oF to p2=5 atm. Employing the
ideal gas model with constant specific heat ratio k, determine the
work and heat transfer per unit mass, in Btu/lb, if (a) n=1.3, (b) n=k .
Evaluate k at T1.
• 3.115 One kilogram of nitrogen fills the cylinder of a piston-cylinder
assembly, as shown in Fig. P3.115. There is no friction between the
piston and the cylinder walls, and the surroundings are at 1 atm. The
initial volume and pressure in the cylinder are 1 m3 and 1 atm,
respectively. Heat transfer to the nitrogen occurs until the volume is
doubled. Determine the heat transfer for the process, in kJ,
assuming the specific heat ratio is constant, k=1.4.
3.135 A piston-cylinder assembly contains air modeled as an ideal
gas with a constant specific heat ratio, k=1.4. The air undergoes a
power cycle consisting of four processes in series:
Process 1–2: Constant-temperature expansion at 600 K from p1=0.5
MPa to p2=0.4 MPa.
Process 2–3: Polytropic expansion with n=k to p3=0.3 MPa.
Process 3–4: Constant-pressure compression to V4=V1.
Process 4–1: Constant-volume heating.
Sketch the cycle on a p–v diagram. Determine (a) the work and heat
transfer for each process, in kJ/kg, and (d) the thermal efficiency.

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 views: 411 posted: 11/25/2011 language: English pages: 22